Advances in Operator Theory

Dominated orthogonally additive operators in lattice-normed spaces

Nariman Abasov and Marat Pliev

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Abstract

‎In this paper, we introduce a new class of operators in lattice-normed spaces‎. ‎We say that an orthogonally additive operator $T$ from a lattice-normed space $(V,E)$ to a lattice-normed space‎ ‎$(W,F)$ is dominated‎, ‎if there exists a positive orthogonally additive operator $S$ from $E$ to $F$ such that $\vert Tx \vert \leq S \vert x \vert$ for any element $x$ of $(V,E)$‎. ‎We show that under some mild‎ ‎conditions‎, ‎a dominated orthogonally additive operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated orthogonally additive operator‎. ‎In the last part of the‎ ‎paper we consider laterally-to-order continuous operators‎. ‎We prove that a dominated orthogonally additive operator is laterally-to-order continuous if and only if the same is its exact dominant‎.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 251-264.

Dates
Received: 27 April 2018
Accepted: 2 August 2018
First available in Project Euclid: 20 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1537408975

Digital Object Identifier
doi:10.15352/aot.1804-1354

Mathematical Reviews number (MathSciNet)
MR3867344

Zentralblatt MATH identifier
06946453

Subjects
Primary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]
Secondary: 47H99: None of the above, but in this section

Keywords
Lattice-normed space‎ ‎vector lattice‎‎ ‎orthogonally additive operator‎ ‎dominated $\mathcal{P}$-operator ‎exact dominant‎ ‎‎laterally-to-order continuous operator‎

Citation

Abasov, Nariman; Pliev, Marat. Dominated orthogonally additive operators in lattice-normed spaces. Adv. Oper. Theory 4 (2019), no. 1, 251--264. doi:10.15352/aot.1804-1354. https://projecteuclid.org/euclid.aot/1537408975


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